Partial Fractions

Partial Fractions
A way of "breaking apart" fractions with polynomials in them.
What are Partial Fractions?
We can do this directly:

Like this (read Using Rational Expressions to learn more):
2
+
3
=
2·(x+1) + (x-2)·3

x-2

x+1

(x-2)(x+1) Which can then be simplified to: =
2x+2 + 3x-6 =
5x-4

x2+x-2x-2

x2-x-2 ... but how do we go in the opposite direction?

That is what we discover here:
How to find the "parts" that make the single fraction
(the "partial fractions").
Why Would We Want To?
First of all ... why would we want to?
Because the partial fractions are each simpler.
This can help solve the more complicated fraction. For example it is very useful in Integral Calculus.
Partial Fraction Decomposition
So let me show you how to do it.
The method is called "Partial Fraction Decomposition", and goes like this:

Step 1: Factor the bottom.

Step 2: Write one partial fraction for each of those factors

Step 3: Multiply through by the bottom so we no longer have fractions
Step 4: Now find the constants!
Substituting the roots ("zeros") of the bottom can help:

And we have our answer:

That was easy! ... almost too easy ...
... because it can be a lot harder!
Now we go into detail on each step.
Proper Rational Expressions
Firstly, this only works for Proper Rational Expressions, where the degree of the top is less thanthe bottom.
The degree is the largest exponent the variable has.
Proper: the degree of the top is less than the degree of the bottom.
Proper:

degree of top is 1 degree of bottom is 3
Improper: the degree of the top is greater than, or equal to, the degree of the bottom.
Improper:

degree of top is 2 degree of bottom is 1
If your expression is Improper, then do polynomial long division first.
Factoring the Bottom
It is up to you to factor the bottom polynomial. See Factoring in Algebra.
But don't factor it into complex numbers ... you may need to stop some factors at quadratic (called